Fractals

Tree Fractal

This fractal actually works just they way one would expect, and prints in just a few seconds. For each branch, a line of length L is drawn, then another branch of length L/2 is drawn at 45° and 135° to the original. These new branches of course generate their own branches of length L/4, which generate L/8, and so on.

Triangle Fractal

The triangle fractal is built in almost the way that it looks; an earlier version actually was built in the way it looks. Essentially, to draw the fractal, three sub-fractals are made of half the size as the original, one in each corner of an equilateral triangle. This is repeated many times, until the size is smaller than the smallest dot that the printer can display. Once there, instead of drawing smaller sub-fractals, it actually draws a triangle between the middle of the lines of the triangle. This creates the same image as it would if it drew the triangle for each fractal (ie if it first drew a large triangle, then a smaller one in each corner-fractal, then nine smaller, and so on), but runs faster and has better precision.

Mandelbrot: in colour

Here are some print times and estimated print times, on the LaserWriter II NT; modern printers are much faster.

IterationsResolutionTime
10036 dpi25 minutes
200144 dpi10 hours
3003003 days (estimated)

The Mandelbrot set is a mathematical set of complex numbers, often printed on computer screens or posters. To calculate it, the computer (or Postscript printer) does a bit of mathematical analysis.

For each point in the complex plane between –2 –2i and 2 + 2i, let the complex number of the point be C, and create another complex number called Z, and set it to 0 + 0i (ie, zero).

Now, calculate Z × Z + C, and then make Z equal to this. Repeat.

If, when this is repeated many times, Z does not become very big, then the point C is said to be in the Mandelbrot set, and is coloured black.

Whenever the Mandelbrot set is done in colour, a pixel’s colour indicates how many times the process was repeated before Z for that point became very large. The outside colour may be only after one or two iterations, the next colour after 5, then next ten, and so on. The black part is actually in the set, and ideally takes forever to reach. For my monochrome Poscript Mandelbrot interpreter, a constant number of iterations is used. At higher resolutions is becomes necessary to increase the iterations, otherwise the edges become incorrectly distorted.


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